Mild steel is fine for standard tongs (such as the ones in the video) however some types of tongs, like Brian's hammer tongs require spring steel.
The reason is because they are built to be lightweight and thin but structurally strong, when you are holding a hammer with tongs you dont want any unnecessary weight added to the load. Spring steel does a much better job at resisting bending and thus you can afford to make a lighter, springier pair of tongs with it but for "regular" tongs mild steel is likely preferable.
Also you dont want to cool spring steel tongs in water as you can crack them.